Binary Labelings for Plane Quadrangulations and their Relatives

Stefan Felsner, Clemens Huemer, Sarah Kappes, David Orden

Abstract

Motivated by the bijection between Schnyder labelings of a plane
triangulation and partitions of its inner edges into three trees, we
look for binary labelings for quadrangulations (whose edges can be
partitioned into two trees). Our labeling resembles many of the
properties of Schnyder's one for triangulations: Apart from being in
bijection with tree decompositions, paths in these trees allow to
define the regions of a vertex such that counting faces in them yields
an algorithm for embedding the quadrangulation, in this case on
a 2-book. Furthermore, as Schnyder labelings have been extended to
3-connected plane graphs, we are able to extend our labeling from
quadrangulations to a larger class of 2-connected bipartite graphs.